Geometry: Mensuration and 3D Shapes

At GCSE you apply mensuration to more complex shapes, including arcs, sectors, pyramids, cones, spheres and frustums. You also use Pythagoras' theorem in three dimensions.

---

Arcs, Sectors and Segments

For a circle of radius r with sector angle θ°:

Quantity	Formula
Arc length	L = (θ/360) × 2πr
Sector area	A = (θ/360) × πr²
Segment area	Sector area − Triangle area

Triangle area inside a sector: A = (1/2)r² sin θ (using the sine area formula)

Example: Find the area of a minor segment, r = 10 cm, θ = 80°. Sector area = (80/360) × π × 100 = 69.81 cm² Triangle area = (1/2) × 100 × sin 80° = 49.24 cm² Segment area = 69.81 − 49.24 = 20.57 cm²

---

Volumes of 3D Shapes